This application calculates the dominant eigenvalue of a 2×2 matrix using the Power Method.
The power iteration process is given by:
$$ x_{k+1} = \frac{A x_k}{\| A x_k \|} $$
where \( A \) is the given matrix and \( x_k \) is the eigenvector approximation at step \( k \).
The Power Method is an iterative technique used to find the dominant eigenvalue (the eigenvalue with the greatest absolute value) of a matrix \( A \). Starting with an initial guess vector \( x_0 \) (here chosen as [1,1]), the method repeatedly applies the matrix \( A \) and normalizes the result:
$$ x_{k+1} = \frac{A x_k}{\| A x_k \|} $$
As \( k \) increases, the vector \( x_k \) converges to the eigenvector corresponding to the dominant eigenvalue \( \lambda \), which is estimated by:
$$ \lambda \approx \frac{(A x_k)_i}{(x_k)_i} $$
In this simulator, after performing the iteration, the final approximate dominant eigenvalue is displayed as text, and the sequence of normalized vectors is visualized on the canvas as arrows originating from the origin. The coordinate axes are drawn and forced to include the origin (0,0) if it lies within the data range.
All formulas are rendered using MathJax.